To give a criterion for the integrability of Banach-Lie
triple systems, we follow the construction of the period group
of a Lie algebra and define the period group of a Lie triple
system as an analogous concept. We show that a Lie triple system
is integrable if and only if its period group is discrete. Along
the way, we see how to turn the path and the loop space of a
pointed symmetric space into pointed symmetric spaces.